Data

• Growth rate values of salmonella in broth
• The listeria.csv file has three columns (Temp, GR sqrtGR) and 29 observations
  • Temp is the temperature used to measure the growth rate
  • GR is the growth rate
  • sqrtGR is the square root of the growth rate
• Lets read the listeria.csv with the read.csv() function and save it as data frame (df)

df <- read.csv("../data/listeria.csv", sep=";", header=TRUE)
head(df)

  Temp      GR sqrtGR
1  0.7 0.00000 0.0000
2  3.4 0.00034 0.0185
3  7.6 0.00127 0.0356
4  9.6 0.00166 0.0407
5 11.2 0.00230 0.0480
6 12.8 0.00256 0.0506

Data

• Lets, check the dataset using the structure (str()) function

str(df)

'data.frame':   28 obs. of  3 variables:
 $ Temp  : num  0.7 3.4 7.6 9.6 11.2 12.8 14.1 15.4 16.9 18.2 ...
 $ GR    : num  0 0.00034 0.00127 0.00166 0.0023 0.00256 0.00311 0.00423 0.00452 0.00587 ...
 $ sqrtGR: num  0 0.0185 0.0356 0.0407 0.048 0.0506 0.0558 0.065 0.0672 0.0766 ...

• We confirm that the data set is composed of 3 variables (columns)
  • Temp: temperature
  • GR: growth rate
  • sqrtGR: square root of the growth rate

Check the data shape

• Check the data by plotting the sqrtGR against Temperature
  • We can observe a non-linear relationship

plot(GR ~ Temp, data=df,
     xlab="Temperature (ºC)", ylab="Growth rate")

plot(sqrtGR ~ Temp, data=df,
     xlab="Temperature (ºC)", ylab="Growth rate")

Secondary models

• Describe the influence of environmental factors on microbial growth rate
• Can be generally classified as:
  • Arrhenius-type models
  • Belehradek-type models
  • Square-root models
  • Cardinal parameter models
• When adjusting a secondary model, using directly the untransformed growth rate response variable produces heterocedastic models
• To obtain an adequate fitting, the growth rate values are often transformed by taking the natural logarithm or the square root

Software

• We use the R software
  • Its free
  • Developed for statistical analysis and plots
  • Functions to fit non-linear models to experimental data
    • nls() function from the```stats package
    • gsl_nls() from gslnls package (GNU Scientific Library)
    • Others
• Download and install the R software: https://cran.r-project.org/
• Install the predmicror package: https://fsqanalytics.github.io/predmicror/

Software

• Install the predmicror package from GitHub repository

devtools::install_github("fsqanalytics/predmicror")

• Load the predmicror package

library(predmicror)

Selected model: Cardinal model for temperature

• We will use the Cardinal model for temperature implemented in the predmicror package
  • https://fsqanalytics.github.io/predmicror/reference/CMTI.html

library(predmicror)
CMTI

function (x, Tmax, Tmin, MUopt, Topt) 
{
    if (!requireNamespace("gslnls", quietly = TRUE)) {
        stop("Package \"gslnls\" must be installed to use this function.", 
            call. = FALSE)
    }
    CMT <- ifelse(x <= Tmin | x >= Tmax, 0, MUopt * (((x - Tmax) * 
        (x - Tmin)^2)/((Topt - Tmin) * ((Topt - Tmin) * (x - 
        Topt) - (Topt - Tmax) * (Topt + Tmin - 2 * x)))))
    result <- sqrt(CMT)
    return(result)
}
<bytecode: 0x57af77b9bab0>
<environment: namespace:predmicror>

Starting values

• To fit non-linear models we need to we need to supply starting values for the model parameters
• So, lets start by defining the starting values

start.values = list(Tmax=47, Tmin=0.1, MUopt=0.020, Topt=37)

Fit using gsl_nls() function

• Now we can fit the CMTI model to the experimental data

library(predmicror)
library(gslnls)
fit <- gsl_nls(sqrtGR ~ CMTI(Temp,Tmax,Tmin,MUopt,Topt),
           data=df,
           start =  start.values
           )
fit

Nonlinear regression model
  model: sqrtGR ~ CMTI(Temp, Tmax, Tmin, MUopt, Topt)
   data: df
    Tmax     Tmin    MUopt     Topt 
44.27284  1.38405  0.02139 37.33238 
 residual sum-of-squares: 0.000484

Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr)

Number of iterations to convergence: 6 
Achieved convergence tolerance: 3.619e-09

Check fitting results

• Now we can summarise and inspect the fitting results by using the summary() function

summary(fit)

Formula: sqrtGR ~ CMTI(Temp, Tmax, Tmin, MUopt, Topt)

Parameters:
       Estimate Std. Error t value Pr(>|t|)    
Tmax  4.427e+01  6.121e-02 723.351   <2e-16 ***
Tmin  1.384e+00  5.564e-01   2.488   0.0202 *  
MUopt 2.139e-02  4.531e-04  47.211   <2e-16 ***
Topt  3.733e+01  3.157e-01 118.245   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.004491 on 24 degrees of freedom

Number of iterations to convergence: 6 
Achieved convergence tolerance: 3.619e-09

• We can extract the model coefficients using the coef() function

coefs <- coef(fit)
coefs

       Tmax        Tmin       MUopt        Topt 
44.27284300  1.38405426  0.02139205 37.33238243 

Parameters confidence intervals

• Extract the confidence intervals of the parameters using the confint() function

confint(fit)

        2.5 %      97.5 %
1 44.14652164 44.39916436
2  0.23575472  2.53235380
3  0.02045686  0.02232723
4 36.68076914 37.98399571

Plot the fitted values

• Define a vector with auxiliary temperature data

new.temp=seq(0.7,44, by=0.2)

• Use the predict() function to compute the prediction interval

fits <- predict(fit,
                newdata = data.frame(Temp=new.temp),
                interval = "prediction", level = 0.95)
str(fits)

 num [1:217, 1:3] 0 0 0 0 0.000574 ...
 - attr(*, "dimnames")=List of 2
  ..$ : NULL
  ..$ : chr [1:3] "fit" "lwr" "upr"

• Check the fits object

head(fits[, ])

              fit          lwr         upr
[1,] 0.0000000000 -0.009268608 0.009268608
[2,] 0.0000000000 -0.009268608 0.009268608
[3,] 0.0000000000 -0.009268608 0.009268608
[4,] 0.0000000000 -0.009268608 0.009268608
[5,] 0.0005735376 -0.008828908 0.009975983
[6,] 0.0015625183 -0.007836776 0.010961813

Plot confidence interval

• Create a plot of the original data with the fitted values superimposed
• Plot the observed data using the plot() function
• Use the lines() function to add the confidence interval

plot(sqrtGR ~ Temp, data=df, ylim=c(0,.2))
lines(new.temp, fits[, 1], col="blue")
lines(new.temp, fits[, 2], col="red")
lines(new.temp, fits[, 3], col="red")

To start

Shiny app CardinalFit: https://ubarron.shinyapps.io/CardinalFit/